Statistical mechanics of three-dimensional magnetohydrodynamics in a multiply connected domain

We study the statistical mechanics of three-dimensional magnetohydrodynamics in a multiply connected domain by constructing a Gibbs ensemble that accounts for the three rugged invariants of the ideal dynamics. The phase space we work with is defined by the eigenfunctions of the curl operator on the space of real three-dimensional solenoidal vector fields, The dynamics in this phase space satisfies the essential Liouville property. The theory predicts the appearance of a steady mean magnetic field-velocity field pair coupled with random fluctuations. It is shown that this mean field-flow satisfies the variational principle delta(E - zeta H - xi K) = 0, where E is the energy, H the magnetic helicity, and K is the cross-helicity. We obtain a meaningful continuum limit in which the magnetic field and velocity field exhibit finite amplitude local fluctuations, while the fluctuations of the vector potential and the velocity stream function vanish, In this limit, the energy and cross-helicity are divided among the mean field-flow and the fluctuations, whereas the fluctuation component of the helicity vanishes, so that the helicity is determined entirely by the mean field. It is shown that, in the continuum limit, the Gibbs ensemble is equivalent to the microcanonical ensemble associated with the conservation of the rugged invariants. Copyright (C) 1998 Elsevier Science B.V.